Inductive Definitions in Bounded Arithmetic: A New Way to Approach P - PowerPoint PPT Presentation

CTFM, Feb 19, 2013, Tokyo Institute of Technology Inductive Definitions in Bounded Arithmetic: A New Way to Approach P vs. PSPACE Naohi Eguchi Mathematical Institute, Tohoku University, Japan

Introduction 1/2 • Purpose in computational complexity: Find limits of realistic computations. • Theoretically: Comparing different notions about computational complexity, e.g. P ̸ =? NP

Introduction 1/2 • Purpose in computational complexity: Find limits of realistic computations. • Theoretically: Comparing different notions about computational complexity, e.g. P ̸ =? NP • Difficult: to compare complexity classes directly. = ⇒ Machine-independent logical approaches. • This talk: new Bounded Arithmetic characterisations of P and PSPACE. (P ⊆ NP ⊆ PSPACE, P ̸ =? PSPACE)

Introduction 2/2 In finite model theory (N. Immermann et al.) 1. P is captured by monotone inductive definitions. 2. PSPACE is captured by non-monotone inductive definitions.

Introduction 2/2 In finite model theory (N. Immermann et al.) 1. P is captured by monotone inductive definitions. 2. PSPACE is captured by non-monotone inductive definitions. Can 1 or 2 be formalised in bounded arithmetic? • to understand what is the most essential principle in P- or PSPACE-computations. • to find new aspects of the relationship between P and PSPACE.

Inductive definition (monotone case) Overview 1/4 Example of inductive definition: N is the smallest set containing 0 closed under x �→ x + 1 .

Inductive definition (monotone case) Overview 1/4 Example of inductive definition: N is the smallest set containing 0 closed under x �→ x + 1 . More precisely: Define an operator F : V → V by x ∈ F ( X ) : ⇔ x = 0 ∨ ∃ y ∈ X ( x = y + 1) .

Inductive definition (monotone case) Overview 1/4 Example of inductive definition: N is the smallest set containing 0 closed under x �→ x + 1 . More precisely: Define an operator F : V → V by x ∈ F ( X ) : ⇔ x = 0 ∨ ∃ y ∈ X ( x = y + 1) . See: • N is the least fixed point of F : F ( N ) ⊆ N , ∀ X ⊆ V [ F ( X ) ⊆ X → N ⊆ X ] • The least fixed point exists since F is monotone: X ⊆ Y ⇒ F ( X ) ⊆ F ( Y ) .

Inductive definition (general case) Overview 2/4 F : V → V ; x ∈ F ( X ) : ⇔ x = 0 ∨ ∃ y ∈ X ( x = y + 1) .  F 0 ∅ :=  F α +1 F ( F α ) := ∪  F γ α<γ F α := ( γ : limit )

Inductive definition (general case) Overview 2/4 F : V → V ; x ∈ F ( X ) : ⇔ x = 0 ∨ ∃ y ∈ X ( x = y + 1) .  F 0 ∅ :=  F α +1 F ( F α ) := ∪  F γ α<γ F α := ( γ : limit ) See: • ∃ α 0 < # P ( V ) such that F α 0 +1 = F ( F α 0 ) = F α 0 . • N = F α 0 .

Inductive definition (finite case) Overview 3/4 F : S → S ( # S < ω ) • There does not always exist m < ω such that F m +1 = F ( F m ) = F m . • However ∃ k ≤ 2 # S , ∃ l > 0 such that ∀ n ≥ l, F k + n = F n .

Inductive definition (finite case) Overview 3/4 F : S → S ( # S < ω ) • There does not always exist m < ω such that F m +1 = F ( F m ) = F m . • However ∃ k ≤ 2 # S , ∃ l > 0 such that ∀ n ≥ l, F k + n = F n .

Inductive definition (finite case) Overview 3/4 F : S → S ( # S < ω ) • There does not always exist m < ω such that F m +1 = F ( F m ) = F m . • However ∃ k ≤ 2 # S , ∃ l > 0 such that ∀ n ≥ l, F k + n = F n . Note: • Choice of k and l is not unique. • But F n plays a role similar to the least fixed point like in infinite case.

Connection to time-complexity Overview 4/4 Suppose: 1. A function f ( x ) is computable in T ( x ) steps. 2. TAPE l denotes the tape description at the l th step in computing f ( x ) ; TAPE 0 = · · · · · · B i 1 i | x | B B ( x = i 1 · · · i | x | (input), i 1 , . . . , i | x | ∈ { 0 , 1 } )

Connection to time-complexity Overview 4/4 Suppose: 1. A function f ( x ) is computable in T ( x ) steps. 2. TAPE l denotes the tape description at the l th step in computing f ( x ) ; TAPE 0 = · · · · · · B i 1 i | x | B B ( x = i 1 · · · i | x | (input), i 1 , . . . , i | x | ∈ { 0 , 1 } )

Connection to time-complexity Overview 4/4 Suppose: 1. A function f ( x ) is computable in T ( x ) steps. 2. TAPE l denotes the tape description at the l th step in computing f ( x ) ; TAPE 0 = · · · · · · B i 1 i | x | B B ( x = i 1 · · · i | x | (input), i 1 , . . . , i | x | ∈ { 0 , 1 } ) Then • TAPE T ( x )+1 = TAPE T ( x ) . • This gives rise to (finite) inductive definition!

Formalising computations 1/2 ⇔ ∃ program to compute f f is computable � �� � Σ 0 1 -formula This gives rise to: Def Let Φ : a set of formulas ⊆ Σ 0 1 & f : a function. f is Φ -definable in T if ∃ A ( ⃗ x, y ) ∈ Φ such that 1. All free variables in A ( ⃗ x, y ) are indicated. m, n ) for ∀ ⃗ m ) ⇔ N | m, n ∈ N . 2. n = f ( ⃗ = A ( ⃗ 3. T ⊢ ∀ ⃗ x ∃ ! yA ( ⃗ x, y ) .

Formalising computations 1/2 ⇔ ∃ program to compute f f is computable � �� � Σ 0 1 -formula This gives rise to: Def Let Φ : a set of formulas ⊆ Σ 0 1 & f : a function. f is Φ -definable in T if ∃ A ( ⃗ x, y ) ∈ Φ such that 1. All free variables in A ( ⃗ x, y ) are indicated. m, n ) for ∀ ⃗ m ) ⇔ N | m, n ∈ N . 2. n = f ( ⃗ = A ( ⃗ 3. T ⊢ ∀ ⃗ x ∃ ! yA ( ⃗ x, y ) .

Formalising computations 2/2 Classical facts: 1. f : primitive recursive ⇔ f : Σ 0 1 -definable in IΣ 1 . (Parsons ’70, Mints ’73, Buss ’86 and Takeuti ’87)

Formalising computations 2/2 Classical facts: 1. f : primitive recursive ⇔ f : Σ 0 1 -definable in IΣ 1 . (Parsons ’70, Mints ’73, Buss ’86 and Takeuti ’87) 2. f ∈ FP ⇔ f : Σ b 1 -definable in S 1 2 . (Buss ’86) • The start of bounded-arithmetic characterisations of complexity classes. Note: By G¨ odel’s incompleteness theorem, not all the computable functions are definable in any reasonable system.

Inductive definitions in 2nd order arithmetic • Inductive definition can be axiomatised in 2nd order arithmetic in the most natural way. Fact 1. Π 1 0 -MID 0 = Π 1 1 -CA 0 . (MID: Monotone Inductive definition) 2. Π 1 0 -MID 0 = Π 0 1 -MID 0 ⊊ Π 0 2 -ID 0 ⊊ Π 0 3 -ID 0 ⊊ · · · .

Inductive definitions in 2nd order arithmetic • Inductive definition can be axiomatised in 2nd order arithmetic in the most natural way. Fact 1. Π 1 0 -MID 0 = Π 1 1 -CA 0 . (MID: Monotone Inductive definition) 2. Π 1 0 -MID 0 = Π 0 1 -MID 0 ⊊ Π 0 2 -ID 0 ⊊ Π 0 3 -ID 0 ⊊ · · · . • Finitary inductive definition can be axiomatised in 2nd order bounded arithmetic.

Foundations of 2nd order bounded arithmetic 1/3 Languages of 2nd order bounded arithmetic: 1. 0 , S , + and · . 2. ⌊ x 2 ⌋ , | x | = ⌈ log 2 ( x + 1) ⌉ and | X | . Importantly x # y = 2 | x |·| y | is not included. Intuition: 1. X, Y, Z · · · ∈ < N { 0 , 1 } . 2. | X | = l if X ≡ i 0 i 1 · · · i l − 1 & i j ∈ { 0 , 1 } . 3. j ∈ X ⇔ i j = 1 if X ≡ i 0 i 1 · · · i l − 1 .

Foundations of 2nd order bounded arithmetic 1/3 Languages of 2nd order bounded arithmetic: 1. 0 , S , + and · . 2. ⌊ x 2 ⌋ , | x | = ⌈ log 2 ( x + 1) ⌉ and | X | . Importantly x # y = 2 | x |·| y | is not included. Intuition: 1. X, Y, Z · · · ∈ < N { 0 , 1 } . 2. | X | = l if X ≡ i 0 i 1 · · · i l − 1 & i j ∈ { 0 , 1 } . 3. j ∈ X ⇔ i j = 1 if X ≡ i 0 i 1 · · · i l − 1 .

Foundations of 2nd order bounded arithmetic 2/3 Def ( Σ B 1 -formulas) 1. Σ B 0 = Π B 0 : the set of formulas containing only bounded number quantifiers ∃ x ≤ t . 2. ∃ ⃗ X ( | ⃗ t ∧ φ ( ⃗ X | ≤ ⃗ X )) ∈ Σ B n +1 if φ ∈ Π B n .

Foundations of 2nd order bounded arithmetic 2/3 Def ( Σ B 1 -formulas) 1. Σ B 0 = Π B 0 : the set of formulas containing only bounded number quantifiers ∃ x ≤ t . 2. ∃ ⃗ X ( | ⃗ t ∧ φ ( ⃗ X | ≤ ⃗ X )) ∈ Σ B n +1 if φ ∈ Π B n . Def (Bit-comprehension axiom) ∀ x ∃ X ≤ x s.t. ∀ j < x ( j ∈ X ↔ φ ( j )) ( ∃ X ≤ x · · · denotes ∃ X ( | X | ≤ x ∧ · · · ) ) Note: ∪ n ∈ N Σ B n ⊆ ∆ 0 1 (exp) ⊆ Σ 0 1 by definition.

Foundations of 2nd order bounded arithmetic 3/3 2nd order arith. 2nd order BA 1st order ob- elements of N ≤ p ( | x | ) jects 2nd order ob- f : N → N f : p ( | x | ) → { 0 , 1 } jects Σ 1 Σ B typical classes n n of formulas ( p : polynomial)

Foundations of 2nd order bounded arithmetic 3/3 2nd order arith. 2nd order BA 1st order ob- elements of N ≤ p ( | x | ) jects 2nd order ob- f : N → N f : p ( | x | ) → { 0 , 1 } jects Σ 1 Σ B typical classes n n of formulas ( p : polynomial) Def V n := BASIC + Σ B n -COMP. Σ B n -COMP: BCA with φ restricted to Σ B n . Thm (Zambella ’96) f ∈ FP Σ P n ⇔ f : Σ B n +1 -definable in V n +1 .

Formalising inductive definitions Def ∀ x, ∃ X ≤ x , ∃ Y ≤ x s.t. Y ̸ = ∅ and 1. ∀ j < x ( P ∅ ϕ ( j ) ↔ j = 0) (i.e. P ∅ ϕ = ∅ ) 2. ∀ Z ∀ j < | Z | ( P S ( Z ) ( j ) ↔ φ ( j, P Z ϕ ) ∧ j < x ) ϕ 3. ∀ j < x ( P X + Y ( j ) ↔ P Y ϕ ( j )) ϕ ( P X ϕ : fresh predicate, S : binary successor X �→ X + 1 ) Recall: 1. F 0 = ∅ 2. F m +1 = F ( F m ) 3. ∃ k ≤ 2 # S , ∃ l ̸ = 0 s.t. F k + l = F l